determine if the tables and graphs below express a direct variation between the variables .if they do the constant of variation and an equation that defines the relation
X 1 2 3 4
Y -3 -6 -9 -12
X -15 10 -20 25
Y -3 2 -4 -5
X 16 20 24 28 32
Y 12 15 18 21 24
Two sets of values are said to have a direct variation, or are said to be directly proportional if y = kx, where k is the constant of variation. It just means that whatever transformation you do to x, will also happen to y. For example, if you double x, y will double.
So to check if the tables have a direct variation, we need to divide each y by x, and get the same quotient.
X 1 2 3 4
Y -3 -6 -9 -12
Quotients: -3/-1 = -3. -6/2 = -3, -9/3 = -3, -12/4 = -3
Since all of them are -3, these express a direct variation, with k (constant of variation) = -3. For the equation, just substitute k, so it’s y = -3x.
X -15 10 -20 25
Y -3 2 -4 -5
Quotients: -3/-15 = 1/5, 2/10 = 1/5 , -4/-20 = 1/5, 5/25 = 1/5
Again, they are all 1/5, so yes to direct variation, k = 1/5. The equation is y =(1/5)x.
X 16 20 24 28 32
Y 12 15 18 21 24
Quotients: 12/16 = 3/4, 15/20 = 3/4, 18/24 = 3/4, 21/28 = 3/4, 24/32 = 3/4
So, yes, they express direct variation, with k = 3/4, and equation y = 3/4 x .
Hope this helps!
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